Riemann Method Research Articles

A new method of solving plane-strain boundary value problems for the double slip and rotation model

Abstract A method of solving plane-strain boundary value problems for a reduced version of the double slip and rotation model is developed. It is assumed that the intrinsic spin vanishes. Elastic strains are neglected. The Mohr–Coulomb yield criterion is adopted. An analogy between the solutions for this model and classical rigid plastic solutions of pressure-independent plasticity is revealed. The method is based on introducing auxiliary variables that satisfy the equation of telegraphy in regions where both families of characteristics are curved. Therefore, Riemann's method can conveniently be applied to solving boundary value problems. The method is employed for analyzing the processes of plane-strain drawing and extrusion through a wedge-shaped die. Friction is neglected. The solution is given in terms of ordinary integrals. The effect of the angle of internal friction on processes’ parameters is revealed. The solution reduces to available solutions of pressure-independent plasticity if the angle of internal friction vanishes.

The Cauchy Problem for the General Telegraph Equation with Variable Coefficients under the Cauchy Conditions on a Curved Line in the Plane

The Riemann method is used to prove the global correctness theorem to Cauchy problem for a general telegraph equation with variable coefficients under Cauchy conditions on a curved line in the plane. The global correctness theorem consists of an explicit Riemann formula for a unique and stable classical solution and a Hadamard correctness criterion for this Cauchy problem. From the formulation of the Cauchy problem, the definition of its classical solutions and the established smoothness criterion of the right-hand side of the equation, its correctness criterion is derived. These results are obtained by Lomovtsev’s new implicit characteristics method which uses only two differential characteristics equations and twelve inversion identities of six implicit mappings. If the righthand side of general telegraph equation depends only on one of two independent variables, then it is necessary and sufficient that it be continuous with respect to this variable. If the right-hand side of this equation depends on two variables and is continuous, then in its integral smoothness requirements it is necessary and sufficient the continuity in one and continuous differentiability in the other variable. The correctness criterion represents the necessary and sufficient smoothness requirements of the right-hand side of the equation and the Cauchy data. From the established global correctness theorem, the well-known Riemann formulas for classical solutions and correctness criteria to Cauchy problems for the general and model telegraph equations in the upper half-plane are derived. In the works of other authors, there is no necessary (minimally sufficient) smoothness on the right-hand sides of the hyperbolic equations of real Cauchy problems for the set of classical (twice continuously differentiable) solutions.

Cross-Sensory EEG Emotion Recognition with Filter Bank Riemannian Feature and Adversarial Domain Adaptation.

Emotion recognition is crucial in understanding human affective states with various applications. Electroencephalography (EEG)-a non-invasive neuroimaging technique that captures brain activity-has gained attention in emotion recognition. However, existing EEG-based emotion recognition systems are limited to specific sensory modalities, hindering their applicability. Our study innovates EEG emotion recognition, offering a comprehensive framework for overcoming sensory-focused limits and cross-sensory challenges. We collected cross-sensory emotion EEG data using multimodal emotion simulations (three sensory modalities: audio/visual/audio-visual with two emotion states: pleasure or unpleasure). The proposed framework-filter bank adversarial domain adaptation Riemann method (FBADR)-leverages filter bank techniques and Riemannian tangent space methods for feature extraction from cross-sensory EEG data. Compared with Riemannian methods, filter bank and adversarial domain adaptation could improve average accuracy by 13.68% and 8.36%, respectively. Comparative analysis of classification results proved that the proposed FBADR framework achieved a state-of-the-art cross-sensory emotion recognition performance and reached an average accuracy of 89.01% ± 5.06%. Moreover, the robustness of the proposed methods could ensure high cross-sensory recognition performance under a signal-to-noise ratio (SNR) ≥ 1 dB. Overall, our study contributes to the EEG-based emotion recognition field by providing a comprehensive framework that overcomes limitations of sensory-oriented approaches and successfully tackles the difficulties of cross-sensory situations.

Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed.

On Some Generalizations of Integral Inequalities in n Independent Variables and Their Applications

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young’s method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities.

Theoretical Analysis of Sheet Beam Electron Gun for Terahertz Vacuum Electron Devices

In the terahertz band, the vacuum electron devices with planar and miniaturized structures require the ultrahigh-precision description of high-density electron beams for formation and transportation with the theoretical analysis and simulation. Thus, the absence of the available theory for the sheet beam electron is put forward to investigate the 3-D spatial potential using a differential geometry and tensor analysis. The equation of electron beam potential can be derived by establishing a moving coordinate frame, carrying out the numerical calculations, and doing series expansion. To solve the complex sheet beam description of the existence of four corner singularities, the Riemann method and conformal mapping are adopted to overcome this difficulty. Then, a 0.22-THz vacuum electron tube using a sheet beam electron gun and corresponding optical system are designed. The beam voltage and current are 16.5 kV and 0.5 A, respectively, with a beam channel size of 1.4 mm <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times0.14$ </tex-math></inline-formula> mm and a uniform magnetic field of 5980G. With the 3-D simulation, the high-density sheet electron beam can transport 18 mm successfully. The spatial potential of numerical calculation with the obtained sheet beam theoretical function and the simulation model are compared to verify the theoretical description. Both the results are with good consistency, which preliminarily proves the effectiveness of the theoretical function. This work will give the deep physical mechanics vision for the development of high-power and high-efficiency sheet beam terahertz tube in the future.

Resolution of challenging problems in quantum cosmology with electromagnetic radiation

We investigate the quantum cosmology of a closed spatially homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) minisuperspace model with electromagnetic radiation as matter content. We solve the corresponding Wheeler–DeWitt equation by utilizing Riemann's zeta function regularization method. We demonstrate that the regularized vacuum energy of the electromagnetic field can overcome factor ordering, boundary conditions, and singularity problems.

Modified Problems for Euler–Darboux Equations with Parameters with Absolute Values Equal to $$ \frac{1}{2} $$

We consider the Euler–Darboux equation with parameters such that their absolute values are equal to \( \frac{1}{2} \). Since the Cauchy problem in the classical formulation is ill-posed for such values of parameters, we propose formulations and solutions of modified Cauchy-type problems with the following values of parameters:(a) \( \alpha =\beta =\frac{1}{2}; \)(b) \( \alpha =-\frac{1}{2}\ \mathrm{and}\ \beta =\frac{1}{2}; \)(c) \( \alpha =\beta =-\frac{1}{2}. \)In case (a), the modified Cauchy problem is solved by the Riemann method. We use the obtained result to formulate the analog of the problem Δ1 in the first quadrant with translated boundary conditions on axes and nonstandard conjunction conditions on the singularity line of the coefficients of the equation y = x. The first condition is gluing normal derivatives of the solution and the second one contains limit values of a combination of the solution and its normal derivatives. The problem is reduced to a uniquely solvable system of integral equations.

Theoretical Analysis and Simulated Verification of Circular Beam Electron Optical System for Terahertz Vacuum Electron Devices

The micro-structure miniaturization of vacuum electron devices (VEDs) in terahertz band requires the high precision description of formation and transportation for the ultrahigh density electron beams with the novelty theoretical analysis and simulation. In this article, the 3-D spatial potential of circular electron beam formation is presented by adopting differential geometry and tensor analysis theory. The specific numerical solution of the circular electron beam potential can be obtained by establishing a moving coordinate frame, carrying out numerical calculation, series expansion with adopting of Riemann method. In order to verify the complex theory, a 0.34 THz vacuum electron tube using thin circular electron beam gun and corresponding optical system are designed and analyzed. The beam voltage and current are 23.8 kV and 0.28 A respectively, with beam channel radius of 0.16 mm and magnetic field of 5250 Gs. Using 3-D simulation of CST, the ultrahigh density electron beam can transport 30 mm with good stability. The spatial potential of numerical calculation with the obtained theoretical functions and the simulation model are compared to verify our theoretical description. Both the results are with good consistent which preliminary proves the effectiveness of the theoretical function. Furthermore, we combine the theoretical function potential with the electron beam envelope under the same coordinate frame, the detailed characteristics of ultrahigh density beam formation with the potential function can be directly understood easily, which will give the deep physical mechanics vision for the development of high power and high-efficiency terahertz tube in the future.

Bilinear auto-Bäcklund transformation, soliton and periodic-wave solutions for a (2+1)-dimensional generalized Kadomtsev–Petviashvili system in fluid mechanics and plasma physics

Fluid mechanics and plasma physics are currently interesting. In fluid mechanics and plasma physics, a (2+1)-dimensional generalized Kadomtsev–Petviashvili system is hereby investigated via symbolic computation. Based on the Hirota method, a bilinear auto-Bäcklund transformation is determined, along with some soliton solutions. Via the Hirota–Riemann method, periodic-wave solutions are derived. We also disscuss the relation between the periodic-wave and soliton solutions, i.e., how the periodic-wave solutions approach the soliton solutions under a limiting procedure. Our results might be useful in understanding certain nonlinear phenomena in fluid mechanics and plasma physics.

Solution of super- and hypersonic gas dynamic problems with a model of high-temperature air

The study considers the solution of a number of problems of supersonic and hypersonic gas dynamics using a model that takes into account the dissociation and ionization of air. The results of verification and validation of the developed numerical method using various difference schemes (the Roe scheme, Rusanov scheme, AUSM scheme) for discretizing convective flows are presented. The formulation of the mathematical model for high-temperature air uses the presence of equilibrium chemical reactions of dissociation and ionization. For this purpose, at high incoming flow velocities, the Kraiko model is applied, which includes equilibrium chemical reactions in air at high temperatures. To discretize the basic equations, the finite volume method on an unstructured grid is applied. One of the features of the constructed mathematical model is the implementation of the transition between physical and conservative variables. Relationships are given, with the help of which the transition from conservative variables to physical ones and vice versa is carried out when using the high-temperature air model. To ensure the stability of numerical calculations, an entropy correction is introduced. The decrease in entropy in the solution of hyperbolic equations is excluded by introducing an artificial viscosity according to Neumann, as well as by using the Godunov method with an exact solution of the Riemann problem and methods based on the approximate solution of the problem of the decay of an arbitrary discontinuity. A number of problems of supersonic gas dynamics (supersonic flow in a channel with a straight step and supersonic flow around a sphere) are numerically solved taking into account high-temperature effects. The criteria for the accuracy of numerical calculations related to the location of shock-wave structures are discussed. The calculated shock-wave structure of the flow is compared with the data available in the literature, as well as with calculations using the perfect gas model. Some results of numerical calculations are compared with the available experimental data. The shock-wave flow patterns obtained in the framework of the inviscid model, which takes into account the effect of viscosity and its dependence on temperature, and the turbulent flow model are compared. On the basis of numerical simulation data, the influence of viscous effects on the flow characteristics in a channel with a straight step and hypersonic flow around a sphere is considered. The influence of various numerical factors on the shape of the bow shock and the presence of fluctuations in the solution behind the shock is emphasized. As part of the work, a computational module was prepared for the commercial package Ansys Fluent, implemented with the help of user programming tools. The prepared module expands the standard capabilities of commercial software focused on solving computational gas dynamics problems, and is available to Ansys Fluent users for solving hypersonic aerodynamics problems. The developed means of numerical simulation can be useful in the design and optimization of hypersonic aircraft.

Generalization of the Riemann method for the trunk gas pipelines considering the quadratic law of resistance

A mathematical model of the problem of pulse propagation in a semi-infinite gas pipeline was developed by expressing the pressure drop by the quadratic law of resistance and the local component of the gas inertia force by the law of conservation of momentum, using the law of conservation of mass in a one-dimensional statement. The model repeats the Riemann problem but takes into account the frictional resistance force. Using an auxiliary function in the form of the natural logarithm of the reduced density, and gauge functions, and certain simplifications, an equation for the reference solution of the problem in terms of gas velocity was derived and solved. For the analytical solution of the problem on gas velocity, the Riemann solution was used, and a refined analytical solution was obtained considering the quadratic law of resistance for the calculation of the perturbed and non-perturbed subdomains.

Plasma slab expansion into vacuum

The problem of collisionless plasma slab expansion into vacuum is solved within a two-temperature hydrodynamic approximation in the dispersionless limit of zero Debye radius. In the framework of such an approach, the solution by the Riemann method provides a quite accurate description of the whole process of plasma dynamics. It is shown that the dispersionless approximation agrees very well with exact numerical solution of the full system of plasma hydrodynamic equations.

Solitons and periodic waves for a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics

Under investigation in this paper is a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics. Soliton and one-periodic-wave solutions are obtained via the Hirota bilinear method and Hirota–Riemann method. Magnitude and velocity of the one soliton are derived. Graphs are presented to discuss the solitons and one-periodic waves: the coefficients in the equation can determine the velocity components of the one soliton, but cannot alter the soliton magnitude; the interaction between the two solitons is elastic; the coefficients in the equation can influence the periods and velocities of the periodic waves. Relation between the one-soliton solution and one-periodic wave solution is investigated.

Solution of Cauchy Problem for the Generalized Gellerstedt Equation

In this paper, the unique solvability of the Cauchy problem for a generalized hyperbolic Gellersted equation with two lines of degeneracy of different order is studied. A modified initial problem is formulated. By using the Riemann method the solution of this problem is constructed in an explicit form inside characteristic triangle. The constructed Riemann function is expressed by special Appell functions and Gauss hypergeometric functions of two variables.

Boundary-value Problems with Data on Characteristics for Hyperbolic Systems of Equations

The main subjects of the present paper are the Goursat and Darboux boundary-value problems for hyperbolic systems with two independent variables. We show that obtained by T.V. Chekmarev in terms of successive approximations formulas for solution of the Goursat problem can be built also by the Riemann method, work out an analog of the Riemann–Hadamard method for the system, and introduce its Riemann–Hadamard matrix. We solve the Darboux problem in terms of the introduced matrix.

Solvability of a Mixed Problem with an Integral Condition for a Third-Order Hyperbolic Equation

In this paper, we examine the solvability of a mixed problem with an integral condition for a third-order equation whose principal part contains the wave operator. The existence and uniqueness of a classical solution to this problem are proved by the Riemann method.

On Formulation of Modified Problems for the Euler-Darboux Equation with Parameters Equalto 1/2 in Absolute Value

We consider the Euler-Darboux equation with parameters equal to 1/2 in absolute value. Since the Cauchy problem in the classical formulation in ill-posed for such values of parameters, we proposeformulations and solutions of modified Cauchy-type problems with the following values of parameters: a)α = β = 1 , b) α = - 1 , β = - 1 , c) α = β = - 1 . In the case а), the modified Cauchy problem is solved2 2 2 2by the Riemann method. We use the obtained result to formulate the analog of the problem Δ1 in the first quadrant with shifted boundary-value conditions on axes and nonstandard conjunction conditions on thesingularity line of the coefficients of the equation y = x. The first condition is gluing normal derivatives of the solution and the second one contains limiting values of combination of the solution and its normal derivatives. The problem is reduced to a uniquely solvable system of integral equations.

Solitons and periodic waves for the (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

Fluid mechanics has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering. Under investigation in this paper is the ($$2+1$$)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. Via the Pfaffian technique and certain constraint on the real constant $$\alpha $$, the Nth-order Pfaffian solutions are derived. One- and two-soliton solutions are obtained via the Nth-order Pfaffian solutions. Based on the Hirota–Riemann method, one- and two-periodic wave solutions are constructed. With the help of the analytic and graphic analysis, we notice that: (1) of the one soliton, amplitude is irrelevant to $$\gamma $$, a real constant coefficient in the equation, velocity along the x direction is independent of $$\gamma $$, while velocity along the y direction is proportional to $$\gamma $$; (2) one soliton keeps its amplitude and velocity invariant during the propagation and total amplitude of the two solitons in the interaction region is lower than that of any soliton; (3) one-periodic wave can be viewed as a superposition of the overlapping solitary waves, placed one period apart; (4) periodic behaviors for the two-periodic wave exist along the x and y directions, respectively; (5) under certain limiting conditions, one-periodic wave solutions approach to the one-soliton solutions and two-periodic wave solutions approach to the two-soliton solutions.

Landau–Khalatnikov Problem in Relativistic Fluid Dynamics

An alternative method is proposed for solving the Landau–Khalatnikov problem of the 1D expansion stage of a hot hadron matter formed during collisions of high-energy particles or atomic nuclei. The solution of relativistic fluid dynamics equations by the Riemann method gives a new representation for the Khalatnikov potential, in which the symmetry of the emerging flow to reflection in the central plane of the initial distribution of the matter is observed. New exact relations that describe the evolution of the energy density at the center of the distribution as well as the motion of boundaries between the general solution and expansion waves are derived. The energy and velocity distributions of particles, including the preexponential factors that have not been considered earlier, are obtained in the Landau approximation.